New derivation of the Lagrangian of a perfect fluid with a barotropic equation of state

Abstract

In this paper we give a simple proof that when the particle number is conserved, the Lagrangian of a barotropic perfect fluid is ${\mathcal{L}}_m=-\rho [c^2+\int P(\rho )/{\rho }^{2}d\rho ]$, where $\rho$ is the \textit{rest mass} density and $P(\rho )$ is the pressure. To prove this result nor additional fields neither Lagrange multipliers are needed. Besides, the result is applicable to a wide range of theories of gravitation. The only assumptions used in the derivation are: 1) the matter part of the Lagrangian does not depend on the derivatives of the metric, and 2) the particle number of the fluid is conserved (${\nabla }_{\sigma }(\rho u^{\sigma })=0$).

References in corpus (13)

• Extra force in $f(R)$ modified theories of gravity
• Action functionals for relativistic perfect fluids
• The IAU 2000 resolutions for astrometry, celestial mechanics and metrology in the relativistic framework: explanatory supplement
• Nonminimal coupling of perfect fluids to curvature
• The Lagrangian description of perfect fluids and modified gravity with an extra force
• Modified gravity with R-matter couplings and (non-)geodesic motion
• The matter Lagrangian and the energy-momentum tensor in modified gravity with non-minimal coupling between matter and geometry
• Parametrized Post-Newtonian Theory of Reference Frames, Multipolar Expansions and Equations of Motion in the N-body Problem
• Relativistic Celestial Mechanics with PPN Parameters
• $\gamma$ Parameter and Solar System constraint in Chameleon Brans Dick Theory
• Stability analysis and Observational Measurement in Chameleonic Generalised Brans--Dicke Cosmology
• A new approach in stability analysis: case study: tachyon cosmology with non-minimally coupled scalar field-matter
• Interacting agegraphic dark energy model in tachyon cosmology coupled to matter

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